Index of Dissimilarity: A Measure of
Relative Ageing of Human Populations
Ashraf Khan Kayani,
HEC Professor of Sociology
Department of Sociology
University of the Punjab, Lahore, Pakistan
Zill-i-Huma
PhD Scholar
Department of Sociology
University of the Punjab, Lahore, Pakistan
Ahmad Raza
Research Fellow
Center for Organization
and Development Studies
University of Management and Technology
Lahore, Pakistan
Abstract:
Index of dissimilarity
is mostly used as a demographic measure of ‘evenness’ of population
distributions. In the present paper, an attempt is made to see if the index can be used as
a measure of relative ageing of human populations. Relative ageing is operationally
defined as a change in the agedness of a closed population resulting from different
population growth rates or the differences in agedness of different age distributions
resulting from growth rates held constant over a long period of time.
For
the present purpose we used nine stable age distributions resulting from varying growth
rates (r = 0.0 through 4.0 with equal interval of 0.5). Using stable age distribution with
r = 2.0
(mid point) as standard, the index of dissimilarity is computed. It is observed that size
of the index does show the differences in ageing of the distributions. However, the size
alone does not show the directionality of the ageing, that is, whether a population is
younger or older. On the other hand, the size alongwith the percentage point differences
between the two distributions is a very useful measure to show the magnitude and
directionality of the agedness of human populations.
Key
words: Index of dissimilarity, relative ageing, human population, agedness
INDEX OF DISSIMILARITY: A MEASURE
OF RELATIVE
AGEING
OF HUMAN POPULATIONS1
Introduction
The present paper
attempts to show that index of dissimilarity can be a useful measure to compare the
“agedness” of the age distributions of human populations in relative terms.
Expressed differently, we show the relationship between the size of the index and agedness
of the age distributions. We also propose that the index alongwith the percent point
difference in the distributions shows the magnitude and directionality of the agedness of
human populations.
Based on the
proportionate distributions and percentage point differences between them, various indices
are used to measure unevenness, concentration or segregation among two or more groups
(Duncan, 1957; Jakubs, 1981; Massay, 1981; White, 1986) The most commonly used index
derived from proportionate distributions is called index of dissimilarity ( ? ).
The index of
dissimilarity has been used by a number of demographers and social scientists (Bogue,
1969; Keyfitz, 1968, Mukherjee, 1973, and Sakoda, 1981) for different purposes. For
example, Keyfitz (1968, p. 47) uses the index of dissimilarity to show how different an
initial distribution of a population is from the stable distribution. Mukherjee (1973)
suggests that the size of ? can be used as a criterion to
determine whether a population is stable or not. Sakado
(1981) generalized the ? so as to apply it to individual groups as well as to provide a single index
for multiple groups.
Methods and Materials
The index of
dissimilarity is calculated by taking two percentage distributions, assuming one as a
standard distribution and subtracting the other distribution from the standard
distribution or vice versa to get a distribution of percentage point differences. The sum
of the categories of alike signs is called index of dissimilarity. One can also take the
total of all percentage point differences irrespective of the signs and divide the total
by 2 to get the index of dissimilarity. Arithmetically:
|
|
 |
1Revised
version of an invited paper presented at 3rd National Conference on
Statistical,
Sciences, Islamic Countries Society of Statistical
Sciences, May 28-29, 2007, Lahore, Pakistan
?
=
Ax _ Bx
2
Where A and B are two
proportionate distributions and X is age category.
To achieve our
objective, we have selected stable age distributions with life expectancy (e?) of 65 years and varying population
growth rates for females. These stable age distributions are taken from Stable Populations
Age Distributions prepared by United Nations (1990) on South Asian Pattern.
We do recognize that
the stable populations are more of an abstract nature than the real ones. However, the
actual age distributions for a country are generally distorted mainly due to emigration
and or immigration. It is for this reason that we selected the stable age distributions
which are closed to migration. Another reason for such a selection is that annual natural
growth rates (r) vary from a minimum of 0.0 % to a maximum of 4.0 %.
Analysis
The selected female
stable age distributions are given in table 1 with growth rates given at the top of each
of the age distributions. Since growth rates decline monotonically (from 4.0 to 0.0) the
nine distributions are monotonically older in the same order – an axiom of stable
population theory.
Table 2.
Index of Dissimilarity and Percentage Point Differences from Standard Age Distribution for
other Stable Distributions, resulting from different Growth Rates
Growth
Rate (%): 0.0
0.5
1.0 1.5
2.0* 2.5 3.0 3.5
4.0
Percentage point
difference of population in indicated age group
Age group
0-1
1-4
5-9
10-14
15-19
20-24
25-29
30-34
35-39
40-44
45-49
50-54
55-59
60-64
65-69
70-74
75+
Index of
Dissimilarity (?)
|
-1.32
-4.56
-4.52
-3.43
-2.44
-1.56
-0.76
-0.05
+0.59
+1.15
+1.63
+2.04
+2.33
+2.48
+2.45
+2.19
+3.77
18.64 |
-1.04
-3.54
-3.45
-2.56
-1.76
-1.06
-0.44
+0.09
+0.56
+0.96
+1.29
+1.56
+1.74
+1.81
+1.75
+1.53
+2.55
13.85 |
-0.72
-2.43
-2.32
-1.68
-1.12
-0.63
-0.21
+0.14
+0.45
+0.7
+0.89
+1.05
+1.14
+1.17
+1.1
+0.95
+1.52
9.11 |
-0.37
-1.25
-1.17
-0.83
-0.53
-0.28
-0.06
+0.11
+0.26
+0.37
+0.46
+0.53
+0.56
+0.56
+0.52
+0.43
+0.69
4.49 |
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 |
+0.39
+1.29
+1.17
+0.77
+0.45
+0.19
-0.02
-0.19
-0.31
-0.4
-0.47
-0.5
-0.51
-0.49
-0.45
-0.37
-0.54
4.25 |
+0.81
+2.61
+2.31
+1.47
+0.81
+0.29
-0.11
-0.43
-0.65
-0.82
-0.93
-0.98
-0.98
-0.93
-0.83
-0.68
-0.96
8.3 |
+1.24
+3.96
+3.42
+2.11
+1.1
+0.32
-0.27
-0.72
-1.03
-1.24
-1.38
-1.42
-1.41
-1.31
-1.16
-0.93
-1.3
12.16 |
+1.68
+5.32
+4.49
+2.67
+1.3
+0.27
-0.49
-1.04
-1.41
-1.66
-1.8
-1.83
-1.78
-1.64
-1.42
-1.13
-1.55
15.74 |
*Standard
Age Distribution
Examination of table 2
suggests that the indexes of dissimilarity for the eight non-standard distributions show a
peculiar pattern. As the growth rate increases or decreases from the standard population
by the same proportion, the index sizes are almost similar; for example, the ? for growth rates 2.5% and 1.5%.
This is further
clarified in Figure 1, where growth rates and the respective indexes of dissimilarity from
the standard age distribution are shown in a scattergram using a linear regression model.
Figure 1: Scattergram of Index of Dissimilarity and
Population Growth Rate (%)
Discussion
The analysis suggests
that the ? alone can only show the difference between the two distributions in age but
does not give the direction whether the population distribution is older or younger, when
compared to the standard one. However, it does indicate that the agedness of a population
is changed.
To overcome the problem
of directionality in ageing, assistance from distribution of percentage point differences
by age groups should be of importance. The understanding of losses or gains in the
proportions in the age groups leads us to argue that the index of dissimilarity (?) for two different growth levels may be the same but their distributions of
gains and losses may well be entirely different (table 2). In our eight distributions of
percentage point differences (of which ? is an outcome), it
is evident that very high and very low growth rates have the larger values of ? and that the value of ? decreases
monotonically from the highest and the lowest growth rates respectively as the value of r
drifts towards the r of the standard one. This is due to the reason of treating the age
distribution resulting from mid-level growth rates (r=2.0). From the growth rate r= 0.0 to
r=1.5, ? increases monotonically.
Similarly, it increases for distributions resulting from r=2.5 through r=4.0. For almost
identical indexes of dissimilarity, the distributions of gains and losses by age groups
are entirely different. For example, if we deduct the distribution with zero growth rate
from the standard distribution (r = 2.0) all age groups above age 35 years gain the
proportions (older populations). On the contrary, for r = 2.5 age groups above age 25
years lose the proportions (younger population).
This shows that the ? alone does not render full information and that the complete percentage
point difference distribution from which ? is derived from is
necessary to illuminate further the differences between the two distributions.
The stable age
distributions in our example are the eventual effects of the constant growth rates. What
the index of dissimilarity gives us, in fact, is the difference between the effects of the
growth rates in the long-term. A large ? shows that the age
of a population is entirely different from the initial or standard population; a small
value of ? signifies little change in the
age, and a medium size ? shows a moderate change in age of
a population. In other words, the size of ? in any two age
distributions will depict the differences in the ageing of the two populations. To see
whether one population is younger or older than the other, one has to look at the
percentage point differences in the age groups between the two distributions.
Source: United Nations, (1990).
Stable Population Age Distributions. Department of
International Economic and Social Affairs, New York.
ST/ESA/SER.R/98. p. 242
References
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Jakubs, John. (1979) “A Consistent Conceptualization Definition of the
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Keyfitz, Nathan. (1968) Introduction
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Massey, Douglas S. (1981) “Social Class and Ethnic Segregation. A
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